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288                       ANALYTICAL MECHANICS
an acceleration - 7 -r, while the earth imparts to the sui r*
an acceleration equal to 7  
Suppose at any instant we impart to both the sun anc the earth a velocity equal and opposite to that of the sur
and apply an acceleration j.   This will bring the sun t<
rest and keep it at rest, without altering the motion of th< earth relative to the sun.* This reduces the problem o: the motion of the earth to that of a particle moving in i central field of force where the acceleration is
,, N           M       m
or                / (r) =                                           (VII
and                 [F=-M2J,                                   (VIII
where                 M = 7 (-W + m)-
Substituting from equation (VIII)  in equation  (V) w< obtain
for the equation of the orbit.   Let uf = u  j~, then th( equation of the orbit takes the form
t'=0.                                (2;
(to2   ' In order to integrate equation (2) let t; = --,
* The acceleration of a particle relative to another moving particle i found by adding the negative of the acceleration of the second particle t< that of the first.